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volume 4, issue 2, article 31, 2003.

Received 27 February, 2003;

accepted 7 March, 2003.

Communicated by:J. Sándor

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Journal of Inequalities in Pure and Applied Mathematics

ON AN OPEN PROBLEM REGARDING AN INTEGRAL INEQUALITY

S. MAZOUZI AND FENG QI

Department of Mathematics Faculty of Sciences University of Annaba P. O. Box 12, Annaba 23000 ALGERIA.

E-Mail:mazouzi.s@voila.fr

Department of Applied Mathematics and Informatics Jiaozuo Institute of Technology

Jiaozuo City, Henan 454000 The People’s Reupublic of China.

E-Mail:qifeng@jzit.edu.cn

2000c Victoria University ISSN (electronic): 1443-5756 029-03

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On an Open Problem Regarding an Integral Inequality S. Mazouzi and Feng Qi

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J. Ineq. Pure and Appl. Math. 4(2) Art. 31, 2003

http://jipam.vu.edu.au

Abstract

In the article, a functional inequality in abstract spaces is established, which gives a new affirmative answer to an open problem posed by Feng Qi in [9].

Moreover, some integral inequalities and a discrete inequality involving sums are deduced.

2000 Mathematics Subject Classification:Primary 39B62; Secondary 26D15.

Key words: Functional inequality, Integral inequality, Jessen’s inequality.

The second author was supported in part by NNSF (#10001016) of China, SF for the Prominent Youth of Henan Province (#0112000200), SF of Henan Innovation Tal- ents at Universities, NSF of Henan Province (#004051800), Doctor Fund of Jiaozuo Institute of Technology, CHINA

1. Introduction

Under what condition does the inequality

(1.1)

Z b a

[f(x)]^tdx≥ Z b

a

f(x)dx ^t−1

hold fort >1?

This problem was proposed by the second author, F. Qi, in [9] after the following inequality was proved:

(1.2)

Z b a

f(x)n+2

dx≥ Z b

a

f(x)dx n+1

,

where f(x) has continuous derivative of the n-th order on the interval [a, b], f⁽ⁱ⁾(a)≥0for0≤i≤n−1, andf⁽ⁿ⁾(x)≥n!.

In the joint paper [13], K.-W. Yu and F. Qi obtained an answer to the above problem by using the integral version of Jessen’s inequality and a property of convexity: Inequality (1.1) is valid for allf ∈C([a, b])such thatRb

a f(x)dx≥ (b−a)^t−1 for givent >1.

Let[x]denote the greatest integer less than or equal tox,f⁽⁻¹⁾(x) =Rx

a f(s)ds, f⁽⁰⁾(x) =f(x),γ(t) = t(t−1)(t−2)· · ·[t−(n−1)]fort ∈(n, n+ 1], and γ(t) = 1fort < 1, wherenis a positive integer. In [12], N. Towghi provided other sufficient conditions for inequality (1.1) to be valid: If f⁽ⁱ⁾(a) ≥ 0for i≤[t−2]andf^[t−2](x)≥γ(t−1)(x−a)^(t−[t]), thenRb

af(x)dx≥(b−a)^t−1 and inequality (1.1) holds.

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T.K. Pogány in [8], by avoiding the assumptions of differentiability used in [9, 12] and the convexity criteria used in [13], and instead using the classical integral inequalities due to Hölder, Nehari, Barnes and their generalizations by Godunova and Levin, established some inequalities which are generalizations, reversed form, or weighted version of inequality (1.1).

In this paper, by employing a functional inequality introduced in [5], which is an abstract generalization of the classical Jessen’s inequality [10], we further establish the following functional inequality (1.4) from which inequality (1.1), some integral inequality, and an interesting discrete inequality involving sums can be deduced.

Theorem 1.1. LetLbe a linear vector space of real-valued functions, pandq be two real numbers such thatp≥q≥1. Assume thatf andgare two positive functions inLandGis a positive linear form onLsuch that

1. G(g)>0, 2. f gandgf^p ∈ L.

If

(1.3) [G(g)]^p−1 ≤[G(gf)]^p−q,

then

(1.4) [G(gf)]^q ≤G(gf^p).

The new inequality (1.4) has the feature that it is stated for summable functions defined on a finite measure space(E,Σ, µ)whoseL¹-norms are bounded from below by some constant involving the measure of the whole space E as well as the exponentspandq.

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2. Lemma and Proof of Theorem 1.1

To prove our main result, Theorem 1.1, it is necessary to recall a functional inequality from [5], which can be stated as follows.

Lemma 2.1. LetLbe a linear vector space of real valued functions andf, g∈ L withg ≥ 0. Assume thatF is a positive linear form onLandϕ :R → Ris a convex function such that

1. F(g) = 1,

2. f gand(ϕ◦f)g ∈ L.

Then

(2.1) ϕ(F(f g))≤F((ϕ◦f)g).

Notice that Lemma2.1 is in fact a form of the classical Jessen inequality.

There is a vast literature on this subject, see, e.g., [1, 2, 3, 4,7, 11] and references therein.

Proof of Theorem1.1. Define a positive linear formF(u) = ^G(u)_G(g), then, we ob- viously have F(g) = 1. From Lemma 2.1, if we take as a convex function ϕ(x) =x^p forp≥1, then

(2.2) [F(gf)]^p ≤F(gf^p),

that is,

G(gf) G(g)

p

≤ G(gf^p) G(g)

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which gives

[G(gf)]^p−q

[G(g)]^p−1 [G(gf)]^q ≤G(gf^p).

Since inequality (1.3) holds, thus inequality (1.4) follows.

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3. Corollaries and Remarks

As a new positve and concrete answer to F. Qi’s problem mentioned at the be- ginning of this paper, we get the following

Corollary 3.1. Let (E,Σ, µ)be a finite measure space and let Lbe the space of all integrable functions on E. If p and q are two real numbers such that p≥q ≥1, andf andg are two positive functions ofLsuch that

1. R

Egdµ >0, 2. f gandgf^p ∈ L, then

(3.1)

Z

E

gf dµ q

≤ Z

E

gf^pdµ,

provided that R

Egf dµp−q

≥ R

Egdµp−1

.

Proof. This follows from Theorem1.1 by takingG(u) =R

Eudµas a positive linear form.

Remark 3.1. We observe that ifp = qandG(g) ≤ 1, then inequality (1.3) is always fulfilled, and accordingly, we have

[G(gf)]^p ≤G(gf^p) for allp≥1.

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Remark 3.2. IfLcontains the constant functions, then for

(3.2) f =







0, p≥q≥1,

[G(g)]^p−q, p > q ≥1, 1, p=q,G(g) = 1,

equality occurs in (1.4)

Remark 3.3. In fact, inequality (1.4) holds even if inequality (1.3), as merely a sufficient condition, is not satisfied. Let p > q ≥ 1, m = ^q−1_p−q and c = q ^p−q_p−1q−11/(p−q)

. If E = [a, b] is a finite interval of R and f(x) =c(x− a)^m, then Rb

a f dxq

= Rb

a f^pdx. On the other hand, inequality (1.3) is no longer satisfied if q ^p−q_p−1p−1

< 1. This is due to the fact that Rb

a f dxp−q

= q ^p−q_p−1^p−1

b−a^p−1 .

Corollary 3.2. Letf ∈L¹(a, b), the space of integrable functions on the inter- val (a, b) with respect to the Lebesgue measure, such that |f(x)| ≥ k(x)a.e.

forx∈(a, b), where

(3.3) (b−a)(p−1)/(p−q)≤ Z b

a

k(x)dx <∞

for somep > q ≥1. Then

(3.4)

Z b a

|f(x)|dx q

≤ Z b

a

|f(x)|^pdx.

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Proof. This follows easily from Lemma2.1.

We now apply Corollary3.2to deduce F. Qi’s main result, Proposition 1.3 in [9], in detail.

Corollary 3.3. Suppose thatf ∈Cⁿ([a, b])satisfiesf⁽ⁱ⁾(a)≥0andf⁽ⁿ⁾(x)≥ n!forx∈[a, b], where0≤i≤n−1andn∈N, the set of all positive integers, then

(3.5)

Z b a

f(x)n+2

dx≥ Z b

a

f(x)dx n+1

.

Proof. Sincef⁽ⁿ⁾(x)≥n!, then successive integrations over[a, x]give f^(n−k)(x)≥ n!

k!(x−a)^k, k = 0,1, . . . , n−1, hence

(x−a)^n−kf^(n−k)(x)≥ n!

k!(x−a)ⁿ, k = 0,1, . . . , n−1.

On the other hand, Taylor’s expansion applied tofwith Lagrange remainder states that

f(x) = f(a) +

n−1

X

k=0

f^(k)(a)

k! (x−a)^k+f⁽ⁿ⁾(ξ)

n! (x−a)ⁿ

≥

n

X

k=0

n!

k!(n−k)!(x−a)ⁿ

= 2ⁿ(x−a)ⁿ,

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whereξ∈(a, x). But sincexis arbitrary and2ⁿ≥n+ 1for alln ∈N, then f(x)≥(n+ 1)(x−a)ⁿ≥0

for allx∈(a, b). Therefore Z b

a

f(x)dx≥(b−a)ⁿ⁺¹,

and inequality (3.5) follows by virtue of Corollary3.2.

Remark 3.4. The function

f : [a, b]→R⁺, x7→f(x) = (x−a)ⁿ⁺¹ (n+ 2)ⁿ

for a fixedn∈Nsatisfiesf ∈Cⁿ([a, b])andf⁽ⁱ⁾(a)≥0, for0≤i≤n−1,but f⁽ⁿ⁾(x) = _(n+2)^(n+1)!n(x−a)forx∈[a, b]. This means that the conditionf⁽ⁿ⁾ ≥n!

on[a, b]is no longer fulfilled. However, we have Z b

a

f dx n+2

= Z b

a

fⁿ⁺³dx= (b−a)⁽ⁿ⁺²⁾² (n+ 2)^(n+1)(n+2). Finally, let us apply Corollary3.1to derive a discrete inequality.

Corollary 3.4. LetE = {a₁, . . . , a_N},f : E → R⁺defined byf(a_i) = b_i for i= 1, . . . , N, and letµbe a discrete positive measure given byµ({a_i}) = α_i >

0fori= 1, . . . , N. If, forp≥q≥1

(3.6)

N

X

i=1

α_i

!^p−1

≤

N

X

i=1

α_ib_i

!^p−q ,

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then we have

(3.7)

N

X

i=1

α_ib_i

!q

≤

N

X

i=1

α_ib^p_i.

If, in particular,α₁ =· · ·=α_N =c >0satisfies

(3.8) c^q−1 ≤ 1

N^p−1

N

X

i=1

b_i

!p−q

,

then

(3.9)

N

X

i=1

b_i

!q

≤ 1 c^q−1

N

X

i=1

b^p_i.

Proof. We observe that

Z

E

f dµ ^p−q

=

N

X

i=1

f(a_i)µ({a_i})

!p−q

=

N

X

i=1

α_ib_i

!^p−q

≥

N

X

i=1

α_i

!p−1

≡[µ(E)]^p−1.

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and thus, the sufficient condition is satisfied. We conclude by Corollary3.1that Z

E

f dµ q

=

N

X

i=1

αibi

!q

≤ Z

E

f^pdµ=

N

X

i=1

αib^p_i.

The proof of inequality (3.9) is a particular case of the above argument, and thus we leave it to the interested reader.

Remark 3.5. The draft version of this paper is available online at http://rgmia.vu.edu.au/v6n1.html. See [6].

Acknowledgements

The authors are indebted to Professor Jozsef Sándor and the anonymous referees for their many helpful comments and for many valuable additions to the list of references.

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References

[1] P.R. BEESACK AND J.E. PE ˇCARI ´C, On Jessen’s inequality for convex functions, J. Math. Anal. Appl. 110 (1985), 536–552.

[2] P.R. BEESACK AND J.E. PE ˇCARI ´C, On Jessen’s inequality for convex functions, II, J. Math. Anal. Appl. 118 (1986), 125–144.

[3] P.R. BEESACK AND J.E. PE ˇCARI ´C, On Jessen’s inequality for convex functions, III, J. Math. Anal. Appl. 156 (1991), 231–239.

[4] S.S. DRAGOMIR, C.E.M. PEARCE AND J.E. PE ˇCARI ´C, On Jessen’s and related inequalities for isotonic sublinear functionals, Acta Sci. Math (Szeged) 61(1995), 373–382.

[5] S. MAZOUZI, A functional inequality, Magh. Math. Rev. 3 (1994), no. 1, 83–87.

[6] S. MAZOUZI AND FENG QI, On an open problem by F. Qi regarding an integral inequality, RGMIA Res. Rep. Coll. 6 (2003), no. 1, Art. 6.

Available online athttp://rgmia.vu.edu.au/v6n1.html.

[7] J.E. PE ˇCARI ´C AND I. RA ¸SA, On Jessen’s inequality, Acta Sci. Math.

(Szeged), 56 (1992), 305–309.

[8] T.K. POGÁNY, On an open problem of F. Qi, J. Inequal.

Pure Appl. Math., 31 (2002), Art. 4. Available online at http://jipam.vu.edu.au/v3n4/016_01.html.

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[9] FENG QI, Several integral inequalities, J. Inequal. Pure Appl. Math., 1(2) (2000), Art. 19. Available online at http://jipam.vu.edu.au/v1n2/001_00.html. RGMIA Res. Rep. Coll. 2(7) (1999), Art. 9, 1039–1042. Available online at http://rgmia.vu.edu.au/v2n7.html.

[10] W. RUDIN, Real and Complex Analysis, McGraw-Hill, Inc., New York, 1974.

[11] J. SÁNDOR, On the Jessen-Hadamard inequality, Studia Univ. Babes- Bolyai Math., 36(1) (1991), 9–15.

[12] N. TOWGHI, Notes on integral inequalities, RGMIA Res.

Rep. Coll., 4(2) (2001), Art. 12, 277–278. Available online at http://rgmia.vu.edu.au/v4n2.html.

[13] KIT-WING YU AND FENG QI, A short note on an integral inequality, RGMIA Res. Rep. Coll., 4(1) (2001), Art. 4, 23–25. Available online at http://rgmia.vu.edu.au/v4n1.html.